In this appendix, we briefly derive the bilinear rogue waves presented in Theorem 1. These new rogue wave expressions can be obtained by applying a new parameterization developed in Ref. [1]} to the bilinear derivation of rogue waves in Ref. [2]}. Specifically, instead of the previous choice (3.11) for the matrix element \(m_{ij}^{(n)}\)  in Ref. [2]}, which we denote as \(\phi _{ij}^{(n)}\)  in this paper, we now choose
\( \phi _{ij}^{(n)}=\left. \frac{1}{i!}(p\partial _p)^i \frac{1}{j!}(q\partial _q)^j \phi ^{(n)}\right|_{p=q=1},\) 
